Archimedes

Archimedes

Overview
Archimedes of Syracuse (Greek
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...

: ; c. 287 BC – c. 212 BC) was a Greek mathematician
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

, physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

, engineer
Engineer
An engineer is a professional practitioner of engineering, concerned with applying scientific knowledge, mathematics and ingenuity to develop solutions for technical problems. Engineers design materials, structures, machines and systems while considering the limitations imposed by practicality,...

, inventor, and astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

. Although few details of his life are known, he is regarded as one of the leading scientist
Scientist
A scientist in a broad sense is one engaging in a systematic activity to acquire knowledge. In a more restricted sense, a scientist is an individual who uses the scientific method. The person may be an expert in one or more areas of science. This article focuses on the more restricted use of the word...

s in classical antiquity
Classical antiquity
Classical antiquity is a broad term for a long period of cultural history centered on the Mediterranean Sea, comprising the interlocking civilizations of ancient Greece and ancient Rome, collectively known as the Greco-Roman world...

. Among his advances in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

 are the foundations of hydrostatics
Fluid statics
Fluid statics is the science of fluids at rest, and is a sub-field within fluid mechanics. The term usually refers to the mathematical treatment of the subject. It embraces the study of the conditions under which fluids are at rest in stable equilibrium...

, statics
Statics
Statics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity...

 and an explanation of the principle of the lever
Lever
In physics, a lever is a rigid object that is used with an appropriate fulcrum or pivot point to either multiply the mechanical force that can be applied to another object or resistance force , or multiply the distance and speed at which the opposite end of the rigid object travels.This leverage...

. He is credited with designing innovative machine
Machine
A machine manages power to accomplish a task, examples include, a mechanical system, a computing system, an electronic system, and a molecular machine. In common usage, the meaning is that of a device having parts that perform or assist in performing any type of work...

s, including siege engines and the screw pump
Archimedes' screw
The Archimedes' screw, also called the Archimedean screw or screwpump, is a machine historically used for transferring water from a low-lying body of water into irrigation ditches...

 that bears his name.
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Quotations

Μη μου τους κύκλους τάραττε! (Me mou tous kyklous taratte!)

Noli turbare circulos meos. or Noli me tangere|Noli tangere circulos meos. (Latin translations)
Encyclopedia
Archimedes of Syracuse (Greek
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...

: ; c. 287 BC – c. 212 BC) was a Greek mathematician
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

, physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

, engineer
Engineer
An engineer is a professional practitioner of engineering, concerned with applying scientific knowledge, mathematics and ingenuity to develop solutions for technical problems. Engineers design materials, structures, machines and systems while considering the limitations imposed by practicality,...

, inventor, and astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

. Although few details of his life are known, he is regarded as one of the leading scientist
Scientist
A scientist in a broad sense is one engaging in a systematic activity to acquire knowledge. In a more restricted sense, a scientist is an individual who uses the scientific method. The person may be an expert in one or more areas of science. This article focuses on the more restricted use of the word...

s in classical antiquity
Classical antiquity
Classical antiquity is a broad term for a long period of cultural history centered on the Mediterranean Sea, comprising the interlocking civilizations of ancient Greece and ancient Rome, collectively known as the Greco-Roman world...

. Among his advances in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

 are the foundations of hydrostatics
Fluid statics
Fluid statics is the science of fluids at rest, and is a sub-field within fluid mechanics. The term usually refers to the mathematical treatment of the subject. It embraces the study of the conditions under which fluids are at rest in stable equilibrium...

, statics
Statics
Statics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity...

 and an explanation of the principle of the lever
Lever
In physics, a lever is a rigid object that is used with an appropriate fulcrum or pivot point to either multiply the mechanical force that can be applied to another object or resistance force , or multiply the distance and speed at which the opposite end of the rigid object travels.This leverage...

. He is credited with designing innovative machine
Machine
A machine manages power to accomplish a task, examples include, a mechanical system, a computing system, an electronic system, and a molecular machine. In common usage, the meaning is that of a device having parts that perform or assist in performing any type of work...

s, including siege engines and the screw pump
Archimedes' screw
The Archimedes' screw, also called the Archimedean screw or screwpump, is a machine historically used for transferring water from a low-lying body of water into irrigation ditches...

 that bears his name. Modern experiments have tested claims that Archimedes designed machines capable of lifting attacking ships out of the water and setting ships on fire using an array of mirrors.

Archimedes is generally considered to be the greatest mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 of antiquity and one of the greatest of all time. He used the method of exhaustion
Method of exhaustion
The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...

 to calculate the area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

 under the arc of a parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

 with the summation of an infinite series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

, and gave a remarkably accurate approximation of pi
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

. He also defined the spiral bearing his name, formulae for the volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

s of surfaces of revolution
Surface of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane ....

 and an ingenious system for expressing very large numbers.

Archimedes died during the Siege of Syracuse when he was killed by a Roman
Roman Republic
The Roman Republic was the period of the ancient Roman civilization where the government operated as a republic. It began with the overthrow of the Roman monarchy, traditionally dated around 508 BC, and its replacement by a government headed by two consuls, elected annually by the citizens and...

 soldier despite orders that he should not be harmed. Cicero
Cicero
Marcus Tullius Cicero , was a Roman philosopher, statesman, lawyer, political theorist, and Roman constitutionalist. He came from a wealthy municipal family of the equestrian order, and is widely considered one of Rome's greatest orators and prose stylists.He introduced the Romans to the chief...

 describes visiting the tomb of Archimedes, which was surmounted by a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 inscribe
Inscribe
right|thumb|An inscribed triangle of a circleIn geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "Figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about...

d within a cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

. Archimedes had proven that the sphere has two thirds of the volume and surface area of the cylinder (including the bases of the latter), and regarded this as the greatest of his mathematical achievements.

Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Mathematicians from Alexandria
Alexandria
Alexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...

 read and quoted him, but the first comprehensive compilation was not made until c. 530 AD by Isidore of Miletus
Isidore of Miletus
Isidore of Miletus was one of the two main Byzantine architects that Emperor Justinian I commissioned to design the church of Hagia Sophia in Constantinople from 532-537A.D.-Summary:...

, while commentaries on the works of Archimedes written by Eutocius
Eutocius of Ascalon
Eutocius of Ascalon was a Greek mathematician who wrote commentaries on several Archimedean treatises and on the Apollonian Conics.- Life and work :...

 in the sixth century AD opened them to wider readership for the first time. The relatively few copies of Archimedes' written work that survived through the Middle Ages
Middle Ages
The Middle Ages is a periodization of European history from the 5th century to the 15th century. The Middle Ages follows the fall of the Western Roman Empire in 476 and precedes the Early Modern Era. It is the middle period of a three-period division of Western history: Classic, Medieval and Modern...

 were an influential source of ideas for scientists during the Renaissance
Renaissance
The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...

, while the discovery in 1906 of previously unknown works by Archimedes in the Archimedes Palimpsest
Archimedes Palimpsest
The Archimedes Palimpsest is a palimpsest on parchment in the form of a codex. It originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse and other authors, which was overwritten with a religious text.Archimedes lived in the...

 has provided new insights into how he obtained mathematical results.

Biography


Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony
Colonies in antiquity
Colonies in antiquity were city-states founded from a mother-city—its "metropolis"—, not from a territory-at-large. Bonds between a colony and its metropolis remained often close, and took specific forms...

 in Magna Graecia
Magna Graecia
Magna Græcia is the name of the coastal areas of Southern Italy on the Tarentine Gulf that were extensively colonized by Greek settlers; particularly the Achaean colonies of Tarentum, Crotone, and Sybaris, but also, more loosely, the cities of Cumae and Neapolis to the north...

. The date of birth is based on a statement by the Byzantine Greek
Byzantine Greeks
Byzantine Greeks or Byzantines is a conventional term used by modern historians to refer to the medieval Greek or Hellenised citizens of the Byzantine Empire, centered mainly in Constantinople, the southern Balkans, the Greek islands, Asia Minor , Cyprus and the large urban centres of the Near East...

 historian John Tzetzes
John Tzetzes
John Tzetzes was a Byzantine poet and grammarian, known to have lived at Constantinople during the 12th century.Tzetzes was Georgian on his mother's side...

 that Archimedes lived for 75 years. In The Sand Reckoner
The Sand Reckoner
The Sand Reckoner is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe according to the then-current model, and invent a way to talk about extremely...

, Archimedes gives his father's name as Phidias, an astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

 about whom nothing is known. Plutarch
Plutarch
Plutarch then named, on his becoming a Roman citizen, Lucius Mestrius Plutarchus , c. 46 – 120 AD, was a Greek historian, biographer, essayist, and Middle Platonist known primarily for his Parallel Lives and Moralia...

 wrote in his Parallel Lives
Parallel Lives
Plutarch's Lives of the Noble Greeks and Romans, commonly called Parallel Lives or Plutarch's Lives, is a series of biographies of famous men, arranged in tandem to illuminate their common moral virtues or failings, written in the late 1st century...

that Archimedes was related to King Hiero II
Hiero II of Syracuse
Hieron II , king of Syracuse from 270 to 215 BC, was the illegitimate son of a Syracusan noble, Hierocles, who claimed descent from Gelon. He was a former general of Pyrrhus of Epirus and an important figure of the First Punic War....

, the ruler of Syracuse. A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever married or had children. During his youth Archimedes may have studied in Alexandria
Alexandria
Alexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...

, Egypt
Ancient Egypt
Ancient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...

, where Conon of Samos
Conon of Samos
Conon of Samos was a Greek astronomer and mathematician. He is primarily remembered for naming the constellation Coma Berenices.-Life and work:...

 and Eratosthenes of Cyrene
Eratosthenes
Eratosthenes of Cyrene was a Greek mathematician, poet, athlete, geographer, astronomer, and music theorist.He was the first person to use the word "geography" and invented the discipline of geography as we understand it...

 were contemporaries. He referred to Conon of Samos as his friend, while two of his works (The Method of Mechanical Theorems
Archimedes' use of infinitesimals
The Method of Mechanical Theorems is a work by Archimedes which contains the first attested explicit use of infinitesimals. The work was originally thought to be lost, but was rediscovered in the celebrated Archimedes Palimpsest...

and the Cattle Problem
Archimedes' cattle problem
Archimedes' cattle problem is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of the sun god from a given set of restrictions...

) have introductions addressed to Eratosthenes.

Archimedes died c. 212 BC during the Second Punic War
Second Punic War
The Second Punic War, also referred to as The Hannibalic War and The War Against Hannibal, lasted from 218 to 201 BC and involved combatants in the western and eastern Mediterranean. This was the second major war between Carthage and the Roman Republic, with the participation of the Berbers on...

, when Roman forces under General Marcus Claudius Marcellus
Marcus Claudius Marcellus
Marcus Claudius Marcellus , five times elected as consul of the Roman Republic, was an important Roman military leader during the Gallic War of 225 BC and the Second Punic War...

 captured the city of Syracuse after a two-year-long siege
Siege
A siege is a military blockade of a city or fortress with the intent of conquering by attrition or assault. The term derives from sedere, Latin for "to sit". Generally speaking, siege warfare is a form of constant, low intensity conflict characterized by one party holding a strong, static...

. According to the popular account given by Plutarch
Plutarch
Plutarch then named, on his becoming a Roman citizen, Lucius Mestrius Plutarchus , c. 46 – 120 AD, was a Greek historian, biographer, essayist, and Middle Platonist known primarily for his Parallel Lives and Moralia...

, Archimedes was contemplating a mathematical diagram
Mathematical diagram
Mathematical diagrams are diagrams in the field of mathematics, and diagrams using mathematics such as charts and graphs, that are mainly designed to convey mathematical relationships, for example, comparisons over time.- Argand diagram :...

 when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable scientific asset and had ordered that he not be harmed.
The last words attributed to Archimedes are "Do not disturb my circles" , a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is often given in Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

 as "Noli turbare circulos meos," but there is no reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch.

The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 and a cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

 of the same height and diameter. Archimedes had proven that the volume and surface area of the sphere are two thirds that of the cylinder including its bases. In 75 BC, 137 years after his death, the Roman orator Cicero
Cicero
Marcus Tullius Cicero , was a Roman philosopher, statesman, lawyer, political theorist, and Roman constitutionalist. He came from a wealthy municipal family of the equestrian order, and is widely considered one of Rome's greatest orators and prose stylists.He introduced the Romans to the chief...

 was serving as quaestor
Quaestor
A Quaestor was a type of public official in the "Cursus honorum" system who supervised financial affairs. In the Roman Republic a quaestor was an elected official whereas, with the autocratic government of the Roman Empire, quaestors were simply appointed....

 in Sicily
Sicily
Sicily is a region of Italy, and is the largest island in the Mediterranean Sea. Along with the surrounding minor islands, it constitutes an autonomous region of Italy, the Regione Autonoma Siciliana Sicily has a rich and unique culture, especially with regard to the arts, music, literature,...

. He had heard stories about the tomb of Archimedes, but none of the locals was able to give him the location. Eventually he found the tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up, and was able to see the carving and read some of the verses that had been added as an inscription. A tomb discovered in a hotel courtyard in Syracuse in the early 1960s was claimed to be that of Archimedes, but its location today is unknown.

The standard versions of the life of Archimedes were written long after his death by the historians of Ancient Rome. The account of the siege of Syracuse given by Polybius
Polybius
Polybius , Greek ) was a Greek historian of the Hellenistic Period noted for his work, The Histories, which covered the period of 220–146 BC in detail. The work describes in part the rise of the Roman Republic and its gradual domination over Greece...

 in his Universal History was written around seventy years after Archimedes' death, and was used subsequently as a source by Plutarch and Livy
Livy
Titus Livius — known as Livy in English — was a Roman historian who wrote a monumental history of Rome and the Roman people. Ab Urbe Condita Libri, "Chapters from the Foundation of the City," covering the period from the earliest legends of Rome well before the traditional foundation in 753 BC...

. It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built in order to defend the city.

The Golden Crown


The most widely known anecdote
Anecdote
An anecdote is a short and amusing or interesting story about a real incident or person. It may be as brief as the setting and provocation of a bon mot. An anecdote is always presented as based on a real incident involving actual persons, whether famous or not, usually in an identifiable place...

 about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape. According to Vitruvius
Vitruvius
Marcus Vitruvius Pollio was a Roman writer, architect and engineer, active in the 1st century BC. He is best known as the author of the multi-volume work De Architectura ....

, a votive crown
Votive crown
A votive crown is a votive offering in the form of a crown, normally in precious metals and often adorned with jewels. Especially in the Early Middle Ages, they are of a special form, designed to be suspended by chains at an altar, shrine or image...

 for a temple had been made for King Hiero II, who had supplied the pure gold
Gold
Gold is a chemical element with the symbol Au and an atomic number of 79. Gold is a dense, soft, shiny, malleable and ductile metal. Pure gold has a bright yellow color and luster traditionally considered attractive, which it maintains without oxidizing in air or water. Chemically, gold is a...

 to be used, and Archimedes was asked to determine whether some silver
Silver
Silver is a metallic chemical element with the chemical symbol Ag and atomic number 47. A soft, white, lustrous transition metal, it has the highest electrical conductivity of any element and the highest thermal conductivity of any metal...

 had been substituted by the dishonest goldsmith. Archimedes had to solve the problem without damaging the crown, so he could not melt it down into a regularly shaped body in order to calculate its density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

.
While taking a bath, he noticed that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

 of the crown. For practical purposes water is incompressible, so the submerged crown would displace an amount of water equal to its own volume. By dividing the mass of the crown by the volume of water displaced, the density of the crown could be obtained. This density would be lower than that of gold if cheaper and less dense metals had been added. Archimedes then took to the streets naked, so excited by his discovery that he had forgotten to dress, crying "Eureka
Eureka (word)
"Eureka" is an interjection used to celebrate a discovery, a transliteration of a word attributed to Archimedes.-Etymology:The word comes from ancient Greek εὕρηκα heúrēka "I have found ", which is the 1st person singular perfect indicative active of the verb heuriskō "I find"...

!" (Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

: "εὕρηκα!," meaning "I have found it!"). The test was conducted successfully, proving that silver had indeed been mixed in.

The story of the golden crown does not appear in the known works of Archimedes. Moreover, the practicality of the method it describes has been called into question, due to the extreme accuracy with which one would have to measure the water displacement. Archimedes may have instead sought a solution that applied the principle known in hydrostatics
Fluid statics
Fluid statics is the science of fluids at rest, and is a sub-field within fluid mechanics. The term usually refers to the mathematical treatment of the subject. It embraces the study of the conditions under which fluids are at rest in stable equilibrium...

 as Archimedes' Principle
Buoyancy
In physics, buoyancy is a force exerted by a fluid that opposes an object's weight. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus a column of fluid, or an object submerged in the fluid, experiences greater pressure at the bottom of the...

, which he describes in his treatise On Floating Bodies. This principle states that a body immersed in a fluid experiences a buoyant force
Buoyancy
In physics, buoyancy is a force exerted by a fluid that opposes an object's weight. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus a column of fluid, or an object submerged in the fluid, experiences greater pressure at the bottom of the...

 equal to the weight of the fluid it displaces. Using this principle, it would have been possible to compare the density of the golden crown to that of solid gold by balancing the crown on a scale with a gold reference sample, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly. Galileo
Galileo Galilei
Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

 considered it "probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself."

The Archimedes Screw



A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse. The Greek writer Athenaeus of Naucratis
Athenaeus
Athenaeus , of Naucratis in Egypt, Greek rhetorician and grammarian, flourished about the end of the 2nd and beginning of the 3rd century AD...

 described how King Hieron II commissioned Archimedes to design a huge ship, the Syracusia
Syracusia
Syracusia was a ancient Greek ship sometimes claimed to be the largest transport ship of antiquity. It only sailed once, from Syracuse in Sicily to Alexandria in the Ptolemaic Kingdom.-General characteristics:...

, which could be used for luxury travel, carrying supplies, and as a naval warship. The Syracusia is said to have been the largest ship built in classical antiquity. According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a gymnasium
Gymnasium (ancient Greece)
The gymnasium in ancient Greece functioned as a training facility for competitors in public games. It was also a place for socializing and engaging in intellectual pursuits. The name comes from the Ancient Greek term gymnós meaning "naked". Athletes competed in the nude, a practice said to...

 and a temple dedicated to the goddess Aphrodite
Aphrodite
Aphrodite is the Greek goddess of love, beauty, pleasure, and procreation.Her Roman equivalent is the goddess .Historically, her cult in Greece was imported from, or influenced by, the cult of Astarte in Phoenicia....

 among its facilities. Since a ship of this size would leak a considerable amount of water through the hull, the Archimedes screw was purportedly developed in order to remove the bilge water. Archimedes' machine was a device with a revolving screw-shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a body of water into irrigation canals. The Archimedes screw is still in use today for pumping liquids and granulated solids such as coal and grain. The Archimedes screw described in Roman times by Vitruvius
Vitruvius
Marcus Vitruvius Pollio was a Roman writer, architect and engineer, active in the 1st century BC. He is best known as the author of the multi-volume work De Architectura ....

 may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon
Hanging Gardens of Babylon
The Hanging Gardens of Babylon were considered to be one of the greatest Seven Wonders of the Ancient World, and the only one of the Wonders which may in fact have been legendary. They were purportedly built in the ancient city-state of Babylon, near present-day Al Hillah, Babil, in Iraq...

. The world's first seagoing steamship
Steamboat
A steamboat or steamship, sometimes called a steamer, is a ship in which the primary method of propulsion is steam power, typically driving propellers or paddlewheels...

 with a screw propeller
Propeller
A propeller is a type of fan that transmits power by converting rotational motion into thrust. A pressure difference is produced between the forward and rear surfaces of the airfoil-shaped blade, and a fluid is accelerated behind the blade. Propeller dynamics can be modeled by both Bernoulli's...

 was the SS Archimedes
SS Archimedes
SS Archimedes was a steamship built in Britain in 1839. She is notable for being the world's first steamship to be driven by a screw propeller....

, which was launched in 1839 and named in honor of Archimedes and his work on the screw.

The Claw of Archimedes


The Claw of Archimedes
Claw of Archimedes
The Claw of Archimedes was an ancient weapon devised by Archimedes to defend the seaward portion of Syracuse's city wall against amphibious assault...

 is a weapon that he is said to have designed in order to defend the city of Syracuse. Also known as "the ship shaker," the claw consisted of a crane-like arm from which a large metal grappling hook was suspended. When the claw was dropped onto an attacking ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it. There have been modern experiments to test the feasibility of the claw, and in 2005 a television documentary entitled Superweapons of the Ancient World built a version of the claw and concluded that it was a workable device.

The Archimedes Heat Ray


The 2nd century AD author Lucian
Lucian
Lucian of Samosata was a rhetorician and satirist who wrote in the Greek language. He is noted for his witty and scoffing nature.His ethnicity is disputed and is attributed as Assyrian according to Frye and Parpola, and Syrian according to Joseph....

 wrote that during the Siege of Syracuse
Siege of Syracuse (212 BC)
The Siege of Syracuse by the Roman Republic took place in 214-212 BC, at the end of which the Magna Graecia Hellenistic city of Syracuse, located on the east coast of Sicily, fell. The Romans stormed the city after a protracted siege giving them control of the entire island of Sicily. During the...

 (c. 214–212 BC), Archimedes destroyed enemy ships with fire. Centuries later, Anthemius of Tralles
Anthemius of Tralles
Anthemius of Tralles was a Greek professor of Geometry in Constantinople and architect, who collaborated with Isidore of Miletus to build the church of Hagia Sophia by the order of Justinian I. Anthemius came from an educated family, one of five sons of Stephanus of Tralles, a physician...

 mentions burning-glass
Burning-glass
A burning glass or burning lens is a large convex lens that can concentrate the sun's rays onto a small area, heating up the area and thus resulting in ignition of the exposed surface. Burning mirrors achieve a similar effect by using reflecting surfaces to focus the light...

es as Archimedes' weapon. The device, sometimes called the "Archimedes heat ray", was used to focus sunlight onto approaching ships, causing them to catch fire.

This purported weapon has been the subject of ongoing debate about its credibility since the Renaissance. René Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...

 rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes. It has been suggested that a large array of highly polished bronze
Bronze
Bronze is a metal alloy consisting primarily of copper, usually with tin as the main additive. It is hard and brittle, and it was particularly significant in antiquity, so much so that the Bronze Age was named after the metal...

 or copper
Copper
Copper is a chemical element with the symbol Cu and atomic number 29. It is a ductile metal with very high thermal and electrical conductivity. Pure copper is soft and malleable; an exposed surface has a reddish-orange tarnish...

 shields acting as mirrors could have been employed to focus sunlight onto a ship. This would have used the principle of the parabolic reflector
Parabolic reflector
A parabolic reflector is a reflective device used to collect or project energy such as light, sound, or radio waves. Its shape is that of a circular paraboloid, that is, the surface generated by a parabola revolving around its axis...

 in a manner similar to a solar furnace
Solar furnace
A solar furnace is a structure that captures sunlight to produce high temperatures, usually for industry. This is done with a curved mirror that acts as a parabolic reflector, concentrating light onto a focal point...

.

A test of the Archimedes heat ray was carried out in 1973 by the Greek scientist Ioannis Sakkas. The experiment took place at the Skaramagas
Skaramagas
Skaramagkas is a small town in the western part of Athens, Greece. It is part of the municipality of Chaidari. The town has a refinery and a neighboring shipyard and docks to carry oil production to other parts of the Mediterranean and the world. The bay area is one of the most productive...

 naval base outside Athens
Athens
Athens , is the capital and largest city of Greece. Athens dominates the Attica region and is one of the world's oldest cities, as its recorded history spans around 3,400 years. Classical Athens was a powerful city-state...

. On this occasion 70 mirrors were used, each with a copper coating and a size of around five by three feet (1.5 by 1 m). The mirrors were pointed at a plywood of a Roman warship at a distance of around 160 feet (50 m). When the mirrors were focused accurately, the ship burst into flames within a few seconds. The plywood ship had a coating of tar paint, which may have aided combustion.

In October 2005 a group of students from the Massachusetts Institute of Technology
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...

 carried out an experiment with 127 one-foot (30 cm) square mirror tiles, focused on a wooden ship at a range of around 100 feet (30 m). Flames broke out on a patch of the ship, but only after the sky had been cloudless and the ship had remained stationary for around ten minutes. It was concluded that the device was a feasible weapon under these conditions. The MIT group repeated the experiment for the television show MythBusters
MythBusters
MythBusters is a science entertainment TV program created and produced by Beyond Television Productions for the Discovery Channel. The series is screened by numerous international broadcasters, including Discovery Channel Australia, Discovery Channel Latin America, Discovery Channel Canada, Quest...

, using a wooden fishing boat in San Francisco as the target. Again some charring occurred, along with a small amount of flame. In order to catch fire, wood needs to reach its autoignition temperature
Autoignition temperature
The autoignition temperature or kindling point of a substance is the lowest temperature at which it will spontaneously ignite in a normal atmosphere without an external source of ignition, such as a flame or spark. This temperature is required to supply the activation energy needed for combustion...

, which is around 300 °C (570 °F).

When MythBusters broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of "busted" (or failed) because of the length of time and the ideal weather conditions required for combustion to occur. It was also pointed out that since Syracuse faces the sea towards the east, the Roman fleet would have had to attack during the morning for optimal gathering of light by the mirrors. MythBusters also pointed out that conventional weaponry, such as flaming arrows or bolts from a catapult, would have been a far easier way of setting a ship on fire at short distances.

In December 2010, MythBusters again looked at the heat ray story in a special edition featuring Barack Obama
Barack Obama
Barack Hussein Obama II is the 44th and current President of the United States. He is the first African American to hold the office. Obama previously served as a United States Senator from Illinois, from January 2005 until he resigned following his victory in the 2008 presidential election.Born in...

, entitled President's Challenge. Several experiments were carried out, including a large scale test with 500 schoolchildren aiming mirrors at a of a Roman sailing ship 400 feet (120 m) away. In all of the experiments, the sail failed to reach the 210 °C (410 °F) required to catch fire, and the verdict was again "busted". The show concluded that a more likely effect of the mirrors would have been blinding, dazzling, or distracting the crew of the ship.

Other discoveries and inventions


While Archimedes did not invent the lever
Lever
In physics, a lever is a rigid object that is used with an appropriate fulcrum or pivot point to either multiply the mechanical force that can be applied to another object or resistance force , or multiply the distance and speed at which the opposite end of the rigid object travels.This leverage...

, he gave an explanation of the principle involved in his work On the Equilibrium of Planes. Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

, and are sometimes attributed to Archytas
Archytas
Archytas was an Ancient Greek philosopher, mathematician, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder of mathematical mechanics, as well as a good friend of Plato....

. According to Pappus of Alexandria
Pappus of Alexandria
Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...

, Archimedes' work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth." Plutarch describes how Archimedes designed block-and-tackle
Block and tackle
A block and tackle is a system of two or more pulleys with a rope or cable threaded between them, usually used to lift or pull heavy loads.The pulleys are assembled together to form blocks so that one is fixed and one moves with the load...

 pulley
Pulley
A pulley, also called a sheave or a drum, is a mechanism composed of a wheel on an axle or shaft that may have a groove between two flanges around its circumference. A rope, cable, belt, or chain usually runs over the wheel and inside the groove, if present...

 systems, allowing sailors to use the principle of lever
Lever
In physics, a lever is a rigid object that is used with an appropriate fulcrum or pivot point to either multiply the mechanical force that can be applied to another object or resistance force , or multiply the distance and speed at which the opposite end of the rigid object travels.This leverage...

age to lift objects that would otherwise have been too heavy to move. Archimedes has also been credited with improving the power and accuracy of the catapult
Catapult
A catapult is a device used to throw or hurl a projectile a great distance without the aid of explosive devices—particularly various types of ancient and medieval siege engines. Although the catapult has been used since ancient times, it has proven to be one of the most effective mechanisms during...

, and with inventing the odometer
Odometer
An odometer or odograph is an instrument that indicates distance traveled by a vehicle, such as a bicycle or automobile. The device may be electronic, mechanical, or a combination of the two. The word derives from the Greek words hodós and métron...

 during the First Punic War
First Punic War
The First Punic War was the first of three wars fought between Ancient Carthage and the Roman Republic. For 23 years, the two powers struggled for supremacy in the western Mediterranean Sea, primarily on the Mediterranean island of Sicily and its surrounding waters but also to a lesser extent in...

. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled.

Cicero
Cicero
Marcus Tullius Cicero , was a Roman philosopher, statesman, lawyer, political theorist, and Roman constitutionalist. He came from a wealthy municipal family of the equestrian order, and is widely considered one of Rome's greatest orators and prose stylists.He introduced the Romans to the chief...

 (106–43 BC) mentions Archimedes briefly in his dialogue
Dialogue
Dialogue is a literary and theatrical form consisting of a written or spoken conversational exchange between two or more people....

 De re publica
De re publica
De re publica is a dialogue on Roman politics by Cicero, written in six books between 54 and 51 BC. It is written in the format of a Socratic dialogue in which Scipio Africanus Minor takes the role of a wise old man — an obligatory part for the genre...

, which portrays a fictional conversation taking place in 129 BC. After the capture of Syracuse c. 212 BC, General Marcus Claudius Marcellus
Marcus Claudius Marcellus
Marcus Claudius Marcellus , five times elected as consul of the Roman Republic, was an important Roman military leader during the Gallic War of 225 BC and the Second Punic War...

 is said to have taken back to Rome two mechanisms used as aids in astronomy, which showed the motion of the Sun, Moon and five planets. Cicero mentions similar mechanisms designed by Thales of Miletus
Thales
Thales of Miletus was a pre-Socratic Greek philosopher from Miletus in Asia Minor, and one of the Seven Sages of Greece. Many, most notably Aristotle, regard him as the first philosopher in the Greek tradition...

 and Eudoxus of Cnidus
Eudoxus of Cnidus
Eudoxus of Cnidus was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy...

. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus
Gaius Sulpicius Gallus
Gaius Sulpicius Gallus or Galus was a general, statesman and orator of the Roman Republic.Under Lucius Aemilius Paulus, his intimate friend, he commanded the 2nd legion in the campaign against Perseus, king of Macedonia, and gained great reputation for having predicted an eclipse of the moon on the...

 to Lucius Furius Philus
Lucius Furius Philus
Lucius Furius Philus was a consul of ancient Rome in 136 BC. He was a member of the Scipionic circle, and particularly close to Scipio Aemilianus. As consul he was involved with the foedus Mancinum, and offered Mancinus to the Numantines...

, who described it thus:
This is a description of a planetarium
Planetarium
A planetarium is a theatre built primarily for presenting educational and entertaining shows about astronomy and the night sky, or for training in celestial navigation...

 or orrery
Orrery
An orrery is a mechanical device that illustrates the relative positions and motions of the planets and moons in the Solar System in a heliocentric model. Though the Greeks had working planetaria, the first orrery that was a planetarium of the modern era was produced in 1704, and one was presented...

. Pappus of Alexandria
Pappus of Alexandria
Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...

 stated that Archimedes had written a manuscript (now lost) on the construction of these mechanisms entitled . Modern research in this area has been focused on the Antikythera mechanism
Antikythera mechanism
The Antikythera mechanism is an ancient mechanical computer designed to calculate astronomical positions. It was recovered in 1900–1901 from the Antikythera wreck. Its significance and complexity were not understood until decades later. Its time of construction is now estimated between 150 and 100...

, another device from classical antiquity that was probably designed for the same purpose. Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing. This was once thought to have been beyond the range of the technology available in ancient times, but the discovery of the Antikythera mechanism in 1902 has confirmed that devices of this kind were known to the ancient Greeks.

Mathematics


While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch
Plutarch
Plutarch then named, on his becoming a Roman citizen, Lucius Mestrius Plutarchus , c. 46 – 120 AD, was a Greek historian, biographer, essayist, and Middle Platonist known primarily for his Parallel Lives and Moralia...

 wrote: "He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life."
Archimedes was able to use infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

s in a way that is similar to modern integral calculus
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

. Through proof by contradiction (reductio ad absurdum
Reductio ad absurdum
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

), he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion
Method of exhaustion
The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...

, and he employed it to approximate the value of pi. He did this by drawing a larger polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

 outside a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

 and a smaller polygon inside the circle. As the number of sides of the polygon increases, it becomes a more accurate approximation of a circle. When the polygons had 96 sides each, he calculated the lengths of their sides and showed that the value of pi lay between 3 (approximately 3.1429) and 3 (approximately 3.1408), consistent with its actual value of approximately 3.1416. He also proved that the area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

 of a circle was equal to pi multiplied by the square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

 of the radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

 of the circle. In On the Sphere and Cylinder
On the Sphere and Cylinder
On the Sphere and Cylinder is a work that was published by Archimedes in two volumes c. 225 BC. It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so.-Contents:The principal formulae...

, Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. This is the Archimedean property
Archimedean property
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some ordered or normed groups, fields, and other algebraic structures. Roughly speaking, it is the property of having no infinitely large or...

 of real numbers.

In Measurement of a Circle
Measurement of a Circle
Measurement of a Circle is a treatise that consists of three propositions by Archimedes. The treatise is only a fraction of what was a longer work.-Proposition one:Proposition one states:...

, Archimedes gives the value of the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

 of 3 as lying between (approximately 1.7320261) and (approximately 1.7320512). The actual value is approximately 1.7320508, making this a very accurate estimate. He introduced this result without offering any explanation of the method used to obtain it. This aspect of the work of Archimedes caused John Wallis to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results."


In The Quadrature of the Parabola
The Quadrature of the Parabola
The Quadrature of the Parabola is a treatise on geometry, written by Archimedes in the 3rd century BC. Written as a letter to his friend Dositheus, the work presents 24 propositions regarding parabolas, culminating in a proof that the area of a parabolic segment is 4/3 that of a certain inscribed...

, Archimedes proved that the area enclosed by a parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

 and a straight line is times the area of a corresponding inscribed triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

 as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio :


If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant line
Secant line
A secant line of a curve is a line that intersects two points on the curve. The word secant comes from the Latin secare, to cut.It can be used to approximate the tangent to a curve, at some point P...

s, and so on. This proof uses a variation of the series which sums to .

In The Sand Reckoner
The Sand Reckoner
The Sand Reckoner is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe according to the then-current model, and invent a way to talk about extremely...

, Archimedes set out to calculate the number of grains of sand that the universe could contain. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote: "There are some, King Gelo (Gelo II, son of Hiero II
Hiero II of Syracuse
Hieron II , king of Syracuse from 270 to 215 BC, was the illegitimate son of a Syracusan noble, Hierocles, who claimed descent from Gelon. He was a former general of Pyrrhus of Epirus and an important figure of the First Punic War....

), who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited." To solve the problem, Archimedes devised a system of counting based on the myriad
Myriad
Myriad , "numberlesscountless, infinite", is a classical Greek word for the number 10,000. In modern English, the word refers to an unspecified large quantity.-History and usage:...

. The word is from the Greek murias, for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8 vigintillion
Names of large numbers
This article lists and discusses the usage and derivation of names of large numbers, together with their possible extensions.The following table lists those names of large numbers which are found in many English dictionaries and thus have a special claim to being "real words"...

, or 8.

Writings


The works of Archimedes were written in Doric Greek
Doric Greek
Doric or Dorian was a dialect of ancient Greek. Its variants were spoken in the southern and eastern Peloponnese, Crete, Rhodes, some islands in the southern Aegean Sea, some cities on the coasts of Asia Minor, Southern Italy, Sicily, Epirus and Macedon. Together with Northwest Greek, it forms the...

, the dialect of ancient Syracuse. The written work of Archimedes has not survived as well as that of Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

, and seven of his treatises are known to have existed only through references made to them by other authors. Pappus of Alexandria
Pappus of Alexandria
Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...

 mentions On Sphere-Making and another work on polyhedra
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

, while Theon of Alexandria
Theon of Alexandria
Theon was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's Elements and Ptolemy's Handy Tables, as well as writing various commentaries...

 quotes a remark about refraction
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

 from the Catoptrica. During his lifetime, Archimedes made his work known through correspondence with the mathematicians in Alexandria
Alexandria
Alexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...

. The writings of Archimedes were collected by the Byzantine
Byzantine Empire
The Byzantine Empire was the Eastern Roman Empire during the periods of Late Antiquity and the Middle Ages, centred on the capital of Constantinople. Known simply as the Roman Empire or Romania to its inhabitants and neighbours, the Empire was the direct continuation of the Ancient Roman State...

 architect Isidore of Miletus
Isidore of Miletus
Isidore of Miletus was one of the two main Byzantine architects that Emperor Justinian I commissioned to design the church of Hagia Sophia in Constantinople from 532-537A.D.-Summary:...

 (c. 530 AD), while commentaries on the works of Archimedes written by Eutocius
Eutocius of Ascalon
Eutocius of Ascalon was a Greek mathematician who wrote commentaries on several Archimedean treatises and on the Apollonian Conics.- Life and work :...

 in the sixth century AD helped to bring his work a wider audience. Archimedes' work was translated into Arabic by Thābit ibn Qurra
Thabit ibn Qurra
' was a mathematician, physician, astronomer and translator of the Islamic Golden Age.Ibn Qurra made important discoveries in algebra, geometry and astronomy...

 (836–901 AD), and Latin by Gerard of Cremona
Gerard of Cremona
Gerard of Cremona was an Italian translator of Arabic scientific works found in the abandoned Arab libraries of Toledo, Spain....

 (c. 1114–1187 AD). During the Renaissance
Renaissance
The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...

, the Editio Princeps (First Edition) was published in Basel
Basel
Basel or Basle In the national languages of Switzerland the city is also known as Bâle , Basilea and Basilea is Switzerland's third most populous city with about 166,000 inhabitants. Located where the Swiss, French and German borders meet, Basel also has suburbs in France and Germany...

 in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin. Around the year 1586 Galileo Galilei
Galileo Galilei
Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

 invented a hydrostatic balance for weighing metals in air and water after apparently being inspired by the work of Archimedes.

Surviving works


  • On the Equilibrium of Planes (two volumes)
The first book is in fifteen propositions with seven postulates
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

, while the second book is in ten propositions. In this work Archimedes explains the Law of the Lever
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

, stating, "Magnitudes are in equilibrium at distances reciprocally proportional to their weights."
Archimedes uses the principles derived to calculate the areas and centers of gravity
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

 of various geometric figures including triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

s, parallelogram
Parallelogram
In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...

s and parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

s.
  • On the Measurement of a Circle
    Measurement of a Circle
    Measurement of a Circle is a treatise that consists of three propositions by Archimedes. The treatise is only a fraction of what was a longer work.-Proposition one:Proposition one states:...

This is a short work consisting of three propositions. It is written in the form of a correspondence with Dositheus of Pelusium, who was a student of Conon of Samos
Conon of Samos
Conon of Samos was a Greek astronomer and mathematician. He is primarily remembered for naming the constellation Coma Berenices.-Life and work:...

. In Proposition II, Archimedes shows that the value of pi is greater than and less than . The latter figure was used as an approximation of pi throughout the Middle Ages and is still used today when only a rough figure is required.
  • On Spirals
    On Spirals
    On Spirals is a treatise by Archimedes in 225 BC. Although Archimedes did not discover the Archimedean spiral, he employed it in this book to square the circle and trisect an angle.-Preface:...

This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral
Archimedean spiral
The Archimedean spiral is a spiral named after the 3rd century BC Greek mathematician Archimedes. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity...

. It is the locus
Locus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....

 of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

. Equivalently, in polar coordinates
Polar coordinate system
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....

  it can be described by the equation
with real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s and . This is an early example of a mechanical curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

 (a curve traced by a moving point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

) considered by a Greek mathematician.
  • On the Sphere and the Cylinder
    On the Sphere and Cylinder
    On the Sphere and Cylinder is a work that was published by Archimedes in two volumes c. 225 BC. It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so.-Contents:The principal formulae...

    (two volumes)
In this treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 and a circumscribed cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

 of the same height and diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

. The volume is 3 for the sphere, and 23 for the cylinder. The surface area is 42 for the sphere, and 62 for the cylinder (including its two bases), where is the radius of the sphere and cylinder. The sphere has a volume that of the circumscribed cylinder. Similarly, the sphere has an area that of the cylinder (including the bases). A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.
  • On Conoids and Spheroids
This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes of sections
Cross section (geometry)
In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...

 of cones
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

, spheres, and paraboloids.
  • On Floating Bodies (two volumes)
In the first part of this treatise, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes
Eratosthenes
Eratosthenes of Cyrene was a Greek mathematician, poet, athlete, geographer, astronomer, and music theorist.He was the first person to use the word "geography" and invented the discipline of geography as we understand it...

 that the Earth is round. The fluids described by Archimedes are not , since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.

In the second part, he calculates the equilibrium positions of sections of paraboloids. This was probably an idealization of the shapes of ships' hulls. Some of his sections float with the base under water and the summit above water, similar to the way that icebergs float. Archimedes' principle
Archimedes' principle
Archimedes' principle relates buoyancy to displacement. It is named after its discoverer, Archimedes of Syracuse.-Principle:Archimedes' treatise On floating bodies, proposition 5, states that...

 of buoyancy is given in the work, stated as follows:
  • The Quadrature of the Parabola
    The Quadrature of the Parabola
    The Quadrature of the Parabola is a treatise on geometry, written by Archimedes in the 3rd century BC. Written as a letter to his friend Dositheus, the work presents 24 propositions regarding parabolas, culminating in a proof that the area of a parabolic segment is 4/3 that of a certain inscribed...

In this work of 24 propositions addressed to Dositheus, Archimedes proves by two methods that the area enclosed by a parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

 and a straight line is 4/3 multiplied by the area of a triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

 with equal base and height. He achieves this by calculating the value of a geometric series that sums to infinity with the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

 .
  • (O)stomachion
    Ostomachion
    Ostomachion, also known as loculus Archimedius and also as syntomachion, is a mathematical treatise attributed to Archimedes. This work has survived fragmentarily in an Arabic version and in a copy of the original ancient Greek text made in Byzantine times...

This is a dissection puzzle
Dissection puzzle
A dissection puzzle, also called a transformation puzzle or Richter Puzzle, is a tiling puzzle where a solver is given a set of pieces that can be assembled in different ways to produce two or more distinct geometric shapes. The creation of new dissection puzzles is also considered to be a type of...

 similar to a Tangram
Tangram
The tangram is a dissection puzzle consisting of seven flat shapes, called tans, which are put together to form shapes. The objective of the puzzle is to form a specific shape using all seven pieces, which may not overlap...

, and the treatise describing it was found in more complete form in the Archimedes Palimpsest
Archimedes Palimpsest
The Archimedes Palimpsest is a palimpsest on parchment in the form of a codex. It originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse and other authors, which was overwritten with a religious text.Archimedes lived in the...

. Archimedes calculates the areas of the 14 pieces which can be assembled to form a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

. Research published by Dr. Reviel Netz of Stanford University
Stanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...

 in 2003 argued that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square. Dr. Netz calculates that the pieces can be made into a square 17,152 ways. The number of arrangements is 536 when solutions that are equivalent by rotation and reflection have been excluded. The puzzle represents an example of an early problem in combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

.
The origin of the puzzle's name is unclear, and it has been suggested that it is taken from the Ancient Greek
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...

 word for throat or gullet, stomachos . Ausonius
Ausonius
Decimius Magnus Ausonius was a Latin poet and rhetorician, born at Burdigala .-Biography:Decimius Magnus Ausonius was born in Bordeaux in ca. 310. His father was a noted physician of Greek ancestry and his mother was descended on both sides from long-established aristocratic Gallo-Roman families...

 refers to the puzzle as Ostomachion, a Greek compound word formed from the roots of (osteon, bone) and (machē – fight). The puzzle is also known as the Loculus of Archimedes or Archimedes' Box.
  • Archimedes' cattle problem
    Archimedes' cattle problem
    Archimedes' cattle problem is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of the sun god from a given set of restrictions...

This work was discovered by Gotthold Ephraim Lessing
Gotthold Ephraim Lessing
Gotthold Ephraim Lessing was a German writer, philosopher, dramatist, publicist, and art critic, and one of the most outstanding representatives of the Enlightenment era. His plays and theoretical writings substantially influenced the development of German literature...

 in a Greek manuscript consisting of a poem of 44 lines, in the Herzog August Library in Wolfenbüttel
Wolfenbüttel
Wolfenbüttel is a town in Lower Saxony, Germany, located on the Oker river about 13 kilometres south of Brunswick. It is the seat of the District of Wolfenbüttel and of the bishop of the Protestant Lutheran State Church of Brunswick...

, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

 in 1773. It is addressed to Eratosthenes and the mathematicians in Alexandria. Archimedes challenges them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...

s. There is a more difficult version of the problem in which some of the answers are required to be square number
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...

s. This version of the problem was first solved by A. Amthor in 1880, and the answer is a very large number, approximately 7.760271.
  • The Sand Reckoner
    The Sand Reckoner
    The Sand Reckoner is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe according to the then-current model, and invent a way to talk about extremely...

In this treatise, Archimedes counts the number of grains of sand that will fit inside the universe. This book mentions the heliocentric
Heliocentrism
Heliocentrism, or heliocentricism, is the astronomical model in which the Earth and planets revolve around a stationary Sun at the center of the universe. The word comes from the Greek . Historically, heliocentrism was opposed to geocentrism, which placed the Earth at the center...

 theory of the solar system
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

 proposed by Aristarchus of Samos
Aristarchus of Samos
Aristarchus, or more correctly Aristarchos , was a Greek astronomer and mathematician, born on the island of Samos, in Greece. He presented the first known heliocentric model of the solar system, placing the Sun, not the Earth, at the center of the known universe...

, as well as contemporary ideas about the size of the Earth and the distance between various celestial bodies. By using a system of numbers based on powers of the myriad
Myriad
Myriad , "numberlesscountless, infinite", is a classical Greek word for the number 10,000. In modern English, the word refers to an unspecified large quantity.-History and usage:...

, Archimedes concludes that the number of grains of sand required to fill the universe is 8 in modern notation. The introductory letter states that Archimedes' father was an astronomer named Phidias. The Sand Reckoner or Psammites is the only surviving work in which Archimedes discusses his views on astronomy.
  • The Method of Mechanical Theorems
This treatise was thought lost until the discovery of the Archimedes Palimpsest
Archimedes Palimpsest
The Archimedes Palimpsest is a palimpsest on parchment in the form of a codex. It originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse and other authors, which was overwritten with a religious text.Archimedes lived in the...

 in 1906. In this work Archimedes uses infinitesimals
Archimedes' use of infinitesimals
The Method of Mechanical Theorems is a work by Archimedes which contains the first attested explicit use of infinitesimals. The work was originally thought to be lost, but was rediscovered in the celebrated Archimedes Palimpsest...

, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes may have considered this method lacking in formal rigor, so he also used the method of exhaustion
Method of exhaustion
The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...

 to derive the results. As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria
Alexandria
Alexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...

.

Apocryphal works


Archimedes' Book of Lemmas
Book of Lemmas
The Book of Lemmas is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositions on circles.-Translations:...

or Liber Assumptorum is a treatise with fifteen propositions on the nature of circles. The earliest known copy of the text is in Arabic
Arabic language
Arabic is a name applied to the descendants of the Classical Arabic language of the 6th century AD, used most prominently in the Quran, the Islamic Holy Book...

. The scholars T. L. Heath
T. L. Heath
Sir Thomas Little Heath was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College...

 and Marshall Clagett
Marshall Clagett
Marshall Clagett was an American scholar who specialized in the history of science before Galileo, especially Archimedes.-Career:...

 argued that it cannot have been written by Archimedes in its current form, since it quotes Archimedes, suggesting modification by another author. The Lemmas may be based on an earlier work by Archimedes that is now lost.

It has also been claimed that Heron's formula for calculating the area of a triangle from the length of its sides was known to Archimedes. However, the first reliable reference to the formula is given by Heron of Alexandria
Hero of Alexandria
Hero of Alexandria was an ancient Greek mathematician and engineerEnc. Britannica 2007, "Heron of Alexandria" who was active in his native city of Alexandria, Roman Egypt...

 in the 1st century AD.

Archimedes Palimpsest



The foremost document containing the work of Archimedes is the Archimedes Palimpsest
Archimedes Palimpsest
The Archimedes Palimpsest is a palimpsest on parchment in the form of a codex. It originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse and other authors, which was overwritten with a religious text.Archimedes lived in the...

. In 1906, the Danish professor Johan Ludvig Heiberg
Johan Ludvig Heiberg (historian)
Johan Ludvig Heiberg was a Danish philologist and historian. He is best known for his discovery of previously unknown texts in the Archimedes Palimpsest, and for his edition of Euclid's Elements that T. L. Heath translated into English...

 visited Constantinople
Constantinople
Constantinople was the capital of the Roman, Eastern Roman, Byzantine, Latin, and Ottoman Empires. Throughout most of the Middle Ages, Constantinople was Europe's largest and wealthiest city.-Names:...

 and examined a 174-page goatskin parchment of prayers written in the 13th century AD. He discovered that it was a palimpsest
Palimpsest
A palimpsest is a manuscript page from a scroll or book from which the text has been scraped off and which can be used again. The word "palimpsest" comes through Latin palimpsēstus from Ancient Greek παλίμψηστος originally compounded from πάλιν and ψάω literally meaning “scraped...

, a document with text that had been written over an erased older work. Palimpsests were created by scraping the ink from existing works and reusing them, which was a common practice in the Middle Ages as vellum
Vellum
Vellum is mammal skin prepared for writing or printing on, to produce single pages, scrolls, codices or books. It is generally smooth and durable, although there are great variations depending on preparation, the quality of the skin and the type of animal used...

 was expensive. The older works in the palimpsest were identified by scholars as 10th century AD copies of previously unknown treatises by Archimedes. The parchment spent hundreds of years in a monastery library in Constantinople before being sold to a private collector in the 1920s. On October 29, 1998 it was sold at auction to an anonymous buyer for $2 million at Christie's
Christie's
Christie's is an art business and a fine arts auction house.- History :The official company literature states that founder James Christie conducted the first sale in London, England, on 5 December 1766, and the earliest auction catalogue the company retains is from December 1766...

 in New York
New York City
New York is the most populous city in the United States and the center of the New York Metropolitan Area, one of the most populous metropolitan areas in the world. New York exerts a significant impact upon global commerce, finance, media, art, fashion, research, technology, education, and...

. The palimpsest holds seven treatises, including the only surviving copy of On Floating Bodies in the original Greek. It is the only known source of The Method of Mechanical Theorems, referred to by Suidas
Suda
The Suda or Souda is a massive 10th century Byzantine encyclopedia of the ancient Mediterranean world, formerly attributed to an author called Suidas. It is an encyclopedic lexicon, written in Greek, with 30,000 entries, many drawing from ancient sources that have since been lost, and often...

 and thought to have been lost forever. Stomachion was also discovered in the palimpsest, with a more complete analysis of the puzzle than had been found in previous texts. The palimpsest is now stored at the Walters Art Museum
Walters Art Museum
The Walters Art Museum, located in Baltimore, Maryland's Mount Vernon neighborhood, is a public art museum founded in 1934. The museum's collection was amassed substantially by two men, William Thompson Walters , who began serious collecting when he moved to Paris at the outbreak of the American...

 in Baltimore
Baltimore
Baltimore is the largest independent city in the United States and the largest city and cultural center of the US state of Maryland. The city is located in central Maryland along the tidal portion of the Patapsco River, an arm of the Chesapeake Bay. Baltimore is sometimes referred to as Baltimore...

, Maryland
Maryland
Maryland is a U.S. state located in the Mid Atlantic region of the United States, bordering Virginia, West Virginia, and the District of Columbia to its south and west; Pennsylvania to its north; and Delaware to its east...

, where it has been subjected to a range of modern tests including the use of ultraviolet
Ultraviolet
Ultraviolet light is electromagnetic radiation with a wavelength shorter than that of visible light, but longer than X-rays, in the range 10 nm to 400 nm, and energies from 3 eV to 124 eV...

 and light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

 to read the overwritten text.

The treatises in the Archimedes Palimpsest are: On the Equilibrium of Planes, On Spirals, Measurement of a Circle
Measurement of a Circle
Measurement of a Circle is a treatise that consists of three propositions by Archimedes. The treatise is only a fraction of what was a longer work.-Proposition one:Proposition one states:...

, On the Sphere and the Cylinder, On Floating Bodies, The Method of Mechanical Theorems
and Stomachion.

Legacy


There is a crater
Impact crater
In the broadest sense, the term impact crater can be applied to any depression, natural or manmade, resulting from the high velocity impact of a projectile with a larger body...

 on the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

 named Archimedes
Archimedes (crater)
Archimedes is a large lunar impact crater on the eastern edges of the Mare Imbrium. To the south of the crater extends the Montes Archimedes mountainous region. On the southeastern rim is the Palus Putredinis flooded plain, containing a system of rilles named the Rimae Archimedes that extend over...

 (29.7° N, 4.0° W) in his honor, as well as a lunar mountain range, the Montes Archimedes
Montes Archimedes
Montes Archimedes is a mountain range on the Moon. It is named after the crater Archimedes that lies to the north, which in turn has an eponym of the Greek mathematician Archimedes....

 (25.3° N, 4.6° W).

The asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...

 3600 Archimedes
3600 Archimedes
3600 Archimedes is a small main belt asteroid, belonging to the Rafita family. It was discovered by Lyudmila Vasil'evna Zhuravleva in 1978. It is named after Archimedes, the ancient Greek scientist...

 is named after him.

The Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...

 for outstanding achievement in mathematics carries a portrait of Archimedes, along with his proof concerning the sphere and the cylinder. The inscription around the head of Archimedes is a quote attributed to him which reads in Latin: "Transire suum pectus mundoque potiri" (Rise above oneself and grasp the world).

Archimedes has appeared on postage stamps issued by East Germany (1973), Greece
Greece
Greece , officially the Hellenic Republic , and historically Hellas or the Republic of Greece in English, is a country in southeastern Europe....

 (1983), Italy
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...

 (1983), Nicaragua
Nicaragua
Nicaragua is the largest country in the Central American American isthmus, bordered by Honduras to the north and Costa Rica to the south. The country is situated between 11 and 14 degrees north of the Equator in the Northern Hemisphere, which places it entirely within the tropics. The Pacific Ocean...

 (1971), San Marino
San Marino
San Marino, officially the Republic of San Marino , is a state situated on the Italian Peninsula on the eastern side of the Apennine Mountains. It is an enclave surrounded by Italy. Its size is just over with an estimated population of over 30,000. Its capital is the City of San Marino...

 (1982), and Spain
Spain
Spain , officially the Kingdom of Spain languages]] under the European Charter for Regional or Minority Languages. In each of these, Spain's official name is as follows:;;;;;;), is a country and member state of the European Union located in southwestern Europe on the Iberian Peninsula...

 (1963).

The exclamation of Eureka!
Eureka (word)
"Eureka" is an interjection used to celebrate a discovery, a transliteration of a word attributed to Archimedes.-Etymology:The word comes from ancient Greek εὕρηκα heúrēka "I have found ", which is the 1st person singular perfect indicative active of the verb heuriskō "I find"...

 attributed to Archimedes is the state motto of California
California
California is a state located on the West Coast of the United States. It is by far the most populous U.S. state, and the third-largest by land area...

. In this instance the word refers to the discovery of gold near Sutter's Mill
Sutter's Mill
Sutter's Mill was a sawmill owned by 19th century pioneer John Sutter in partnership with James W. Marshall. It was located in Coloma, California, at the bank of the South Fork American River...

 in 1848 which sparked the California Gold Rush
California Gold Rush
The California Gold Rush began on January 24, 1848, when gold was found by James W. Marshall at Sutter's Mill in Coloma, California. The first to hear confirmed information of the gold rush were the people in Oregon, the Sandwich Islands , and Latin America, who were the first to start flocking to...

.

A movement for civic engagement targeting universal access to health care in the US state of Oregon
Oregon
Oregon is a state in the Pacific Northwest region of the United States. It is located on the Pacific coast, with Washington to the north, California to the south, Nevada on the southeast and Idaho to the east. The Columbia and Snake rivers delineate much of Oregon's northern and eastern...

 has been named the "Archimedes Movement," headed by former Oregon Governor John Kitzhaber
John Kitzhaber
John Albert Kitzhaber is the 37th Governor of Oregon. He served as the 35th Governor of Oregon from 1995 to 2003 and became the first person to be elected to the office three times when he was re-elected to a non-consecutive third term in 2010...

.

Further reading


Republished translation of the 1938 study of Archimedes and his works by an historian of science. Complete works of Archimedes in English.

The Works of Archimedes online


External links